Properties

Label 2-2205-1.1-c1-0-28
Degree $2$
Conductor $2205$
Sign $1$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 5-s + 2·10-s − 11-s + 3·13-s − 4·16-s + 3·17-s + 6·19-s + 2·20-s − 2·22-s + 4·23-s + 25-s + 6·26-s + 29-s + 6·31-s − 8·32-s + 6·34-s + 12·38-s − 6·41-s − 6·43-s − 2·44-s + 8·46-s + 9·47-s + 2·50-s + 6·52-s + 10·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s − 0.301·11-s + 0.832·13-s − 16-s + 0.727·17-s + 1.37·19-s + 0.447·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.185·29-s + 1.07·31-s − 1.41·32-s + 1.02·34-s + 1.94·38-s − 0.937·41-s − 0.914·43-s − 0.301·44-s + 1.17·46-s + 1.31·47-s + 0.282·50-s + 0.832·52-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.287238192\)
\(L(\frac12)\) \(\approx\) \(4.287238192\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.050119098227053621632888047745, −8.252623699886366464992405726957, −7.18674908490146366638229016691, −6.50048591700478498626320913392, −5.54795317274704426653960185894, −5.25756591986209044151466901261, −4.20509313138730818892241036590, −3.31721807269730857488302455154, −2.65178092612008457940193320214, −1.20406663212713068028112418239, 1.20406663212713068028112418239, 2.65178092612008457940193320214, 3.31721807269730857488302455154, 4.20509313138730818892241036590, 5.25756591986209044151466901261, 5.54795317274704426653960185894, 6.50048591700478498626320913392, 7.18674908490146366638229016691, 8.252623699886366464992405726957, 9.050119098227053621632888047745

Graph of the $Z$-function along the critical line